A novel finite-time average consensus protocol based on event-triggered nonlinear control strategy for multiagent systems
- Xiaobo Wang^{1}Email author,
- Juelong Li^{1, 2},
- Jianchun Xing^{1} and
- Ronghao Wang^{1}
https://doi.org/10.1186/s13660-017-1533-6
© The Author(s) 2017
Received: 25 May 2017
Accepted: 20 September 2017
Published: 16 October 2017
Abstract
We present a novel finite-time average consensus protocol based on event-triggered control strategy for multiagent systems. The system stability is proved. The lower bound of the interevent time is obtained to guarantee that there is no Zeno behavior. Moreover, the upper bound of the convergence time is obtained. The relationship between the convergence time and protocol parameter with initial state is analyzed. Lastly, simulations are conducted to verify the effectiveness of the results.
Keywords
1 Introduction
In recent years, many applications required a lot of vehicles or robots to work cooperatively and accomplish a complicated task. Given this, many researchers have devoted themselves to the studies of coordination control of multiagent systems [1, 2]. The primary researches in this field include the problems of consensus [3], flocking [4], formation control [5, 6], collective behavior of swarms [7, 8], etc. Of these, the issue of consensus is the basis of studying other problems. Multiagent consensus refers to the design of a proper consensus protocol based on the local information of each agent such that all agents can reach an agreement with regard to certain quantities of interest [4].
In practical multiagent systems, each agent is usually equipped with a small embedded microprocessor and has limited energy, which usually has only limited computing power and working time. These disadvantages drive researchers to develop event-triggered control schemes, and some important achievements have been made recently [9–11]. For example, in [12] the authors introduced the deterministic event-triggered strategy to develop consensus control algorithms, and the lower bound of the interevent time was obtained to guarantee that there is no Zeno behavior. In [13] the problems of event-triggered integrated fault detection, isolation, and control for discrete-time linear systems were considered. It was shown that the amount of data that is sent through the sensor-to-filter and filter-to-actuator channels are dramatically decreased by using an event-triggered technique applied to both the sensor and filter nodes. In [14], the authors proposed a new multiagent consensus event-based control approach. The measurement broadcasts were scheduled in an event-based fashion, and the continuous monitoring of each neighbor’s state is no longer demanded. In [15] the authors proposed a combinational measurement strategy to event design and developed a new event-triggered control algorithm. In this strategy, each agent is only triggered at his own event time, which lowers the frequency of controller updates and reduces the amount of communication. In [16] the authors proposed a self-triggered consensus algorithm for multiagent systems. The algorithm is simpler in formulation and computation. Thus, more energy can be saved using the proposed algorithm in practical multiagent systems.
Moreover, the convergence time is a significant performance indicator for a consensus protocol in the study of the consensus problem. In most works the protocols only achieve state consensus in infinite time interval, that is, the consensus is only achieved asymptotically. However, the stability or performance of multiagent systems in a finite time interval needs to be considered in several cases. The finite-time stability focuses on the behavior of system responses over a finite time interval [17, 18]. Therefore, studying the finite-time stability of multiagent systems is valuable to some degree. The multiagent finite-time stability analysis has also elicited the attention of many researchers [19–22].
Recently, there are few results reported in the literature to address finite-time event-triggered control consensus protocols for multiagent systems. To the best of our knowledge, in [23] the authors presented two novel nonlinear consensus protocols based on event-driven strategy to investigate the finite-time consensus problem of leaderless and leader-following multiagent systems. However, many parameters exist in the proposed protocols, making the protocols complex and restricted, and the relationship between the convergence time and parameters is unclear. Inspired by these, a new consensus protocol based on the event-triggered control strategy is proposed in this paper.
The main contributions of this paper can be summarized as follows: (1) a new finite-time consensus protocol based on the event-triggered control strategy for multiagent systems is presented, and the system stability is proved. The protocol is simpler in formulation and computation. (2) The lower bound of the inter-event time is gotten to guarantee there is no Zeno behavior. (3) The upper bound of convergence time is obtained. The relationship between the convergence time and protocol parameter, the initial state, is analyzed.
The rest of this paper is organized as follows. In Section 2, we introduce some essential background and present the problem statement. The main results and the proof are provided in Section 3. The simulation results are shown in Section 4, and the conclusions and the future works are provided in Section 5.
Notations
\(\mathbf{1}= [1,1,\ldots,1]^{\mathrm{T}}\) with compatible dimensions, \(\mathcal{I}_{n} = \{ 1,2,\ldots,n\}\). \(|x|\) is the absolute value of a real number x, \(\|\cdot\|\) denotes the 2-norm in \(R^{n}\), \(\operatorname{span}\{ \mathbf{1}\} = \{\boldsymbol{\varepsilon} \in R^{n}:\boldsymbol{\varepsilon} = r\mathbf{1},r \in R\}\), and E is the Euler number (approximately 2.71828).
2 Background and problem statement
2.1 Preliminaries
In this subsection, we introduce some basic definitions and results of algebraic graph theory. Comprehensive conclusions on algebraic graph theory are found in [24]. Moreover, we present two essential lemmas.
For an undirected graph \(\mathcal{G} = ( \mathcal {V},\mathcal{E},\mathcal{A} )\) with n vertices, \(\mathcal{V} = \{ v_{1},v_{2},\ldots,v_{n}\}\) is the vertex set, and \(\mathcal{E}\) is the edge set, the adjacency matrix \(\mathcal{A} = [a_{ij}]\) is the \(n \times n\) matrix defined by \(a _{ij}=1\) for \((i,j) \in\mathcal{E}\) and \(a _{ij}=0\) otherwise. The neighbors of a vertex \(v _{i}\) are denoted by \(\mathcal {N}_{i} = \{ v_{j} \in\mathcal{V}:(v_{i},v_{j}) \in \mathcal{E}\}\), and then the vertices \(v _{i}\) and \(v _{j}\) are called adjacent. A path from \(v _{i}\) to \(v _{j}\) is a sequence of distinct vertices starting with \(v _{i}\) and ending with \(v _{j}\) such that consecutive vertices are adjacent. A graph is regarded as connected if there is a path between any two distinct vertices. The Laplacian of a graph \(L= [l _{ij}] \in R^{n \times n}\) is defined by \(l_{ij} = \sum_{k = 1,k \ne i}^{n} a_{ik}\) for \(i=j\) and \(l _{ij}=- a _{ij}\) otherwise. L has always a zero eigenvalue, and 1 is the associated eigenvector. We denote the eigenvalues of L by \(0= \lambda _{1}(L)\leq\lambda_{2}(L)\leq\cdots\leq\lambda _{n}(L)\). For an undirected and connected graph, \(\lambda_{2}(L) = \min_{\boldsymbol{\varepsilon} \ne0,1^{T}\boldsymbol{\varepsilon} = 0}\frac{\boldsymbol{\varepsilon}^{\mathrm{T}}L\boldsymbol{\varepsilon}}{ \boldsymbol{\varepsilon}^{\mathrm{T}}\boldsymbol{\varepsilon}} > 0\). Therefore, if \(\mathbf{1}^{\mathrm{T}} \boldsymbol{\varepsilon} = 0\), then \(\boldsymbol{\varepsilon} ^{\mathrm{T}} L \boldsymbol{\varepsilon} \geq \lambda_{2}(L) \boldsymbol{\varepsilon} ^{\mathrm{T}} \boldsymbol{\varepsilon} \).
Lemma 1
([25])
Lemma 2
([26])
2.2 Problem formulation
With the given protocol \(u _{i}\), for any initial state, if there is a stable equilibrium \(x ^{*}\) and a time \(t ^{*}\) satisfying \(x_{i}= x ^{*}\) for all \(i \in\mathcal{I}_{n}\) with \(t \geq t ^{*}\), then the finite-time consensus problem is solved [25]. In addition, the average consensus problem is solved if the final consensus state is the average value of the initial state, namely, \(x_{i}(t) = \sum_{i = 1}^{n} x_{i} (0) / n\) for all \(i \in\mathcal{I}_{n}\) with \(t \to\infty\).
2.3 Event-triggered control consensus protocol
In the event design, we suppose that each agent can measure its own state \(x _{i}(t)\) and obtain its neighbors’ states stably. For each vertex \(v_{i}\) and \(t \geq 0\), introduce a measurement error \(e _{i}(t)\) and denote the vector \(\mathbf{e} (t)=[e _{1}(t), e _{2}(t),\ldots,e _{n}(t)]\). The time instants at which the events are triggered are defined by the condition \(f(e(t _{i}))=0\). Suppose the triggering time sequence of vertex \(v _{i}\) is \(t^{i} = 0,\tau_{1}, \ldots,\tau_{s}^{i}, \ldots\) .
It is well known that the control algorithm is a piecewise constant function, and the value of the input is equal to the last control update.
3 Main results
3.1 Stability analysis
In this subsection, we study protocol (6). Now, we are in a position to present our main results.
Theorem 1
Proof
Therefore, κ is time invariant.
If \(V(t)=0\), then \(\boldsymbol{\delta} (t)=0\), which implies that \(u _{i} = 0\), \(i \in\mathcal{I}_{n}\). Thus, \({x}(t) \in \operatorname{span}(\mathbf{1})\). Therefore, the system stability is guaranteed, and the novel protocol can solve the finite-time average consensus problem. □
Remark 1
For each agent, an event is triggered as long as the triggered function satisfies \(f_{i} ( t,e_{i}(t),\delta_{i}(t) ) = 0\).
Remark 2
The role of the parameter μ in the triggered function (8) is adjusting the rate of decrease for the Lyapunov function. From equation (8) we know that when the parameter μ is large, the allowable error is large. This means that when μ is large, the trigger frequency is low. From equation (27) we know that when μ is large, the convergence time is long.
3.2 Existence of a lower bound for interevent times
In the event-triggered control conditions, the agent cannot exhibit Zeno behavior. Namely, for any initial, the interevent times \(\{ \tau _{i + 1} - \tau_{i} \}\) defined by equation (8) are lower bounded by a strictly positive time τ. This is proven in the following theorem.
Theorem 2
Proof
Remark 4
From equation (34) it is easy to see that the minimum interevent time increases with μ.
Remark 5
Note that if we set \(\alpha=1\) in protocol (6), then the finite-time nonlinear event-triggered control strategy becomes the typical event-triggered linear consensus protocol studied in [9]. However, the event-triggered linear consensus protocol can only make agents achieve consensus asymptotically, whereas the proposed consensus protocol in this paper can solve the consensus problem in finite time.
3.3 Performance analysis
In this subsection, the relationship between convergence time and other factors, including initial state and parameter α is studied.
Firstly, we study the relationship between convergence time and initial states. In the consensus problem, rather than the size of the initial states, the disagreement between states is more concerned. By definition, \(V(0)\) measures the disagreement of the initial states with final state. From equation (27) we easily see that the convergence time increases as \(V(0)\) increases.
Letting \(dT _{u}( \alpha)/d \alpha=0\), we get \(\alpha= 1 - 2 / \ln ( 4V(0)\lambda_{2}(D) )\). Therefore, given that \(0< \alpha<1\), if \(V(0)\lambda_{2}(D) < E^{2} / 4\), then the convergence time increases as increases; if \(V(0)\lambda_{2}(D) > E^{2} / 4\), then the convergence time decreases initially and then increases as α becomes large, and when \(\alpha= 1 - 2 / \ln ( 4V(0)\lambda_{2}(D) )\), the convergence time gets the minimum value. Generally, the value of \(V(0)\lambda_{2}(D)\) is always large, and to reduce the convergence time, we can set \(\alpha= 1 - 2 / \ln ( 4V(0)\lambda_{2}(D) )\).
Remark 6
Convergence time is defined as the amount of time the system consumes to reach a consensus. The precise convergence time of the studied nonlinear protocol is difficult to obtain. The above conclusions were obtained based on the upper bound of convergence time in equation (27).
4 Simulations
5 Conclusions
- (1)
The proposed protocol can solve the finite-time average consensus problem.
- (2)
The lower bound of the interevent time was obtained to guarantee that there is no Zeno behavior.
- (3)
The larger the difference in the initial state, the longer the convergence time. Moreover, if \(V(0)\lambda_{2}(D) > E^{2} / 4\), then the convergence time decreases initially, then increases as α becomes large, and when \(\alpha= 1 - 2 / \ln ( 4V(0)\lambda_{2}(D) )\), the convergence time gets the minimum value.
In this paper, the authors only considered first-order multiagent systems. Our future works will focus on extending the conclusions to second-order or higher-order multiagent systems with switching topologies, measurement noise, time delays, and so on.
Declarations
Acknowledgements
This project was supported by the National Natural Science Foundation of China (Grant No. 61603414). We would like to express our appreciation to the anonymous referees and the Associate Editor for their valuable comments and suggestions.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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